Sylvester's Problem and Mock Heegner Points

@article{Dasgupta2017SylvestersPA,
  title={Sylvester's Problem and Mock Heegner Points},
  author={Samit Dasgupta and John Voight},
  journal={arXiv: Number Theory},
  year={2017}
}
We prove that if $p \equiv 4,7 \pmod{9}$ is prime and $3$ is not a cube modulo $p$, then both of the equations $x^3+y^3=p$ and $x^3+y^3=p^2$ have a solution with $x,y \in \mathbb{Q}$. 
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